Class 10 - Chapter 4: Quadratic Equations

Section A (1 Mark each)

Q1. Which of the following is a quadratic equation?

(a) $x^2 + 3x + 1 = (x-2)^2$ (b) $(x+1)^2 = 2(x-3)$ (c) $(x+2)(x-1) = x^2 - 2x - 3$ (d) $x^3 - 4x^2 - x + 1 = (x-2)^3$

Q2. The discriminant of the quadratic equation $2x^2 - 4x + 3 = 0$ is:

(a) -8 (b) 8 (c) 40 (d) -40

Q3. The roots of the equation $x^2 - 0.04 = 0$ are:

(a) $\pm 0.02$ (b) $\pm 0.2$ (c) $0.4$ (d) $0.2$

Q4. Assertion (A): The equation $x^2 + 4x + 5 = 0$ has no real roots.
Reason (R): If the discriminant $D < 0$, the quadratic equation has no real roots.

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

Section B (2 Marks each)

Q5. Find the roots of the quadratic equation $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$ by factorization.

Solve on white paper.

Q6. Find the value of $k$ for which the quadratic equation $2x^2 + kx + 3 = 0$ has two equal real roots.

Solve on white paper.

Q7. Check whether the following is a quadratic equation: $(x-2)(x+1) = (x-1)(x+3)$.

Solve on white paper.

Section C (3 Marks each)

Q8. Find two consecutive positive integers, sum of whose squares is 365.

Solve on white paper.

Q9. Solve for $x$: $\frac{1}{x+4} - \frac{1}{x-7} = \frac{11}{30}, x \neq -4, 7$.

Solve on white paper.

Section D (5 Marks)

Q10. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Solve on white paper.

Section E (4 Marks)

Q11. Case Study: Prayer Hall

A charity trust decides to build a prayer hall having a carpet area of 300 sq m with its length one metre more than twice its breadth.

(i) Form a quadratic equation to represent this situation.

(ii) Find the discriminant of the equation formed.

(iii) Find the length and breadth of the prayer hall.

Solve on white paper.